Fluid Mechanics

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??????????????

• Fluid Mechanics

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??????????????(Fluid Mechanics)

• CHAPTER OUTLINE
• ??????? (Pressure)
• ???????????????? ????????????????????
• ?????????????
• ?????????(Buoyant Forces) ??? ??????????????????
  (Archimedess Principle)
• ?????????????? (Fluid Dynamics)
• ????????????????? (Bernoullis Equation)
• ????????????????????????????????

3
????????????(States of Matter)

• ??????? (Solid)
• ????????????????????????????
• ??????? (Liquid)
• ?????????????????????????????????????
• ???? (Gas)
• ???????????????????????????????

4
?????? (Fluids)

• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ??????? (cohesive forces) ????????????????????????
  ??????????????????????????
• ??????????? (liquids) ??? ???? (gases)
  ????????????????

5
??????????????????????????????

• ?????????? (Fluid Statics)
• ????????????????????????????????
• ?????????????????(Fluid Dynamics)
• ?????????????????????????????????????

6
???????????(Forces in Fluids)

• ???????????????????????????? (shearing stresses)
  ???? ???????????(tensile stresses)
• ??????????????????????????????????????????????????
  ????????? ???????????????????????????????????????
  ?
• ??????????????????????????????????????????????????
  ??????????????????

7
???????(Pressure)

• ??????? P ????????????????????? ?
  ???????????????????????????????
  ??????????????????????????????????????????????????
  ?

Definition of pressure
8
???????(Pressure), ???

• ????????????????????????
• ??????????????????????????????????????????????????
  
• ???????????????????????????????????????
  ???????????????? dF ???????????????? dA
  ??????? dF P dA
• ?????????????????? pascal (Pa)

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???????(Pressure) ??? ???(Force)

• ???????????????????????? (scalar)
  ?????????????????????????(vector)
• ??????????????????????????????????????????????????
  ????????????????

10
?????????????(Measuring Pressure)

• ????????????????????????????????????????????
• ??????????????????????????????????????????
  ???????????????????????
• ????????????????????????????????????

11
???????????(snowshoes)

• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ?????????????????????????????????????????????????
  ??????????????????????????????????

12
???????????

• ??????????????????????????????????????????????????
  ???????????????????????????????????????????????
  ?????????????????????????????????????????
• ?????????????????????????????????????????????????
• ?????????????????????????????????

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????????????????????????????????????

• ????????????????????????????????????
• ???????????????????????????????? ??? ?
  ????????????????????????????????(static
  equilibrium)
• ??? ? ??? ??????????????????????????????????????
  
• ????????????????????????????????????????

14
??????????? (Density)

• ?????????????????????????? ???????????????????????
  ??
• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ??????
• ????????????????????????? ? ?????????????????????
  ??????????????????????????????????????????????????
  ?

15
??????????????????????????????
16
?????????????????

• ??????????????????????????????????????????????????
  ?????????????
• ??????????????????????????????????? A
• ????????????????????????????????????? d ???? d
  h
• ?????????????????????????????????????????????????(
  ?????)

17
????????????????? , ???

• ???????????????????????? r
• ????????????????????????????????????? ?
  ?????????????
• ??????????????????????????????
• ?????????????????????????????????????????????????
  
• ??????????????????????????????????????????????
  P0A
• ??????????????????????????????????????????????????
  PA
• ??????????????????????????????? Mg
• ????????????????????????????

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????????????????? , ???????

• ?????????????????????
• ??????????????????????
• ?????????????
• P
• ??????? P ?????????? h ???????????????? P0
  ???????????????? rgh

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???????????????(Atmospheric Pressure)

• ???????????????????????? ??? P0
  ????????????????????????????????? P0
  ??????????????????
• P0 1.00 atm 1.013 x 105 Pa

20
???????????

• ??????????????????????????????????????????? P
  ??????????????????????? 1000 ?????????????????????
  ?? ?????????????????????????? ethyl alcohol
  ???????????????? 806 ???????????????????????
  ?????? ?????????????????????????????
• ???????? P
• ??????? P
• ??????? P
• ??????????

21
???????? ????????????????????(The Force on a
Dam)

• ???????????????????? H ????????????????????? w
  ?????? ???????????????????????????????????????

22
???????? ????????????????????(The Force on a
Dam),???

• ???????????????????? H ????????????????????? w
  ?????? ???????????????????????????????????
• ??????
• ??????????????????????????????????????????????????
  ??????????????????????????????????????( FPA )
  ???????????????????????? dF P dA ????????
• ?????
• ???????????????????????????????????????

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??????????? (Pascals Law)

• ?????????????????????????????????????? P0
• ??????????????????????????????????????????????????
  ??????????????????????????????????????????
• ?????????????????????????????

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??????????? (Pascals Law) , ???

• ???????????????????????????????????????? (Blaise
  Pascal)
• ???????????????????????????????
  ??????????????????????????????????????????????????
  ????? ???????????????????????????

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?????????????????????????

• ?????????????????????
• ?????????????????????? ???????????????????????????
  ??????
• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ??????????

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????????????????????????? , ???

• ??????????????????? ???????
• ???????? ???
  ???????????????????????????????
• ??????????????? W1 W2
• ?????????????????????????? (Conservation of
  Energy)

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?????????????????????????

• ???????????????(Hydraulic brakes)
• ??????????(Car lifts)
• ??????????????(Hydraulic jacks)
• ???????(Forklifts)

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????????

• ???????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ???????? 5.00 ??. ??????????????????????????????
  ???????????? 15.00 ??. ??????????
  ???????????????????????????????????
  ?????????????????????????? 13,300 ??????

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?????????????(Pressure Measurements)
???????????( Barometer)

• ??????????????? Torricelli
• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ?????
• ???????????????????????????????????
• ??????????????????????????? ?????????
• ?????????????? 1 ???????? , 1 atm 0.760 m (???
  Hg)

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?????????????(Pressure Measurements)
???????????(Manometer)

• ??????????????????????????????????????????????????
  ??????
• ???????????????????????? (U-shaped)
  ?????????????????????????????
• ??????????????????????????????????????????????????
  ?????
• ????????????? B ??? P0?gh

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????????????????????????????(Absolute Pressure
and Gauge Pressure)

• P P0 rgh
• P ??? ???????????????
• P P0 ???????? ??????????
• ???????????????? rgh
• ????????????????????????????????????????
  ??????????????

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????????? (Buoyant Force)

• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ??????????????? ???????????????????????

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????????? (Buoyant Force) , ???

• ????????? (buoyant force) ????????????????????????
  ???????????????????????
• ??????? ??????????????????????????????????????????
  ???? ???????????
• ??????????????????????????????????????????????????
  ????

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????????? (Buoyant Force) , ???

• ???????????? (B) ?????????????????????????????????
  ????????????
• ???????????????????????????????????????
• ??????????????????????????????????????????????????
  ?????? ?????????????????? ?????????????????

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????????????????(Archimedess Principle)
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????????????????(Archimedess Principle) , ???

• ??????????????????????????????????????????????????
  ????????????
• ??????????????????????
• Archimedess Principle does not refer to the
  makeup of the object experiencing the buoyant
  force
• The objects composition is not a factor since
  the buoyant force is exerted by the fluid

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????????????????(Archimedess Principle) , ???

• ??????????????????????????????????????????????????
  ?????? Pt A
• ??????????????????????????????????????????????????
  ?????????? Pb A
• B (Pb Pt) A Mg

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???????????????? ?????????????????????????????
??

• ??????????????????????????????????????????????????
  ????????????????
• ?????????????????????????
• ?????????????????????????
  w mg
• ???????????
  B - Fg

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???????????????? ?????????????????????????????
?? , ???

• If the density of the object is less than the
  density of the fluid, the unsupported object
  accelerates upward
• If the density of the object is more than the
  density of the fluid, the unsupported object
  sinks
• The motion of an object in a fluid is determined
  by the densities of the fluid and the object

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Archimedess PrincipleFloating Object

• The object is in static equilibrium
• The upward buoyant force is balanced by the
  downward force of gravity
• Volume of the fluid displaced corresponds to the
  volume of the object beneath the fluid level

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Archimedess PrincipleFloating Object, cont

• The fraction of the volume of a floating object
  that is below the fluid surface is equal to the
  ratio of the density of the object to that of the
  fluid

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Archimedess Principle, Crown Example

• Archimedes was (supposedly) asked, Is the crown
  made of pure gold?
• Crowns weight in air 7.84 N
• Weight in water (submerged) 6.84 N
• Buoyant force will equal the apparent weight loss
• Difference in scale readings will be the buoyant
  force

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Archimedess Principle, Crown Example, cont.

• B Fg T2
• (Weight in air weight in water)
• Archimedess principle says
• B rgV
• Then to find the material of the crown, rcrown
  mcrown in air / V

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Archimedess Principle, Iceberg Example

• What fraction of the iceberg is below water?
• The iceberg is only partially submerged and so
  Vfluid / Vobject applies
• The fraction below the water will be the ratio of
  the volumes (Vwater / Vice)

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Archimedess Principle, Iceberg Example, cont

• Vice is the total volume of the iceberg
• Vwater is the volume of the water displaced
• This will be equal to the volume of the iceberg
  submerged
• About 89 of the ice is below the waters surface

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Types of Fluid Flow Laminar

• Laminar flow
• Steady flow
• Each particle of the fluid follows a smooth path
• The paths of the different particles never cross
  each other
• The path taken by the particles is called a
  streamline

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Types of Fluid Flow Turbulent

• An irregular flow characterized by small
  whirlpool like regions
• Turbulent flow occurs when the particles go above
  some critical speed

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Viscosity

• Characterizes the degree of internal friction in
  the fluid
• This internal friction, viscous force, is
  associated with the resistance that two adjacent
  layers of fluid have to moving relative to each
  other
• It causes part of the kinetic energy of a fluid
  to be converted to internal energy

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Ideal Fluid Flow

• There are four simplifying assumptions made to
  the complex flow of fluids to make the analysis
  easier
• (1) The fluid is nonviscous internal friction
  is neglected
• (2) The flow is steady the velocity of each
  point remains constant

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Ideal Fluid Flow, cont

• (3) The fluid is incompressible the density
  remains constant
• (4) The flow is irrotational the fluid has no
  angular momentum about any point

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Streamlines

• The path the particle takes in steady flow is a
  streamline
• The velocity of the particle is tangent to the
  streamline
• A set of streamlines is called a tube of flow

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Equation of Continuity

• Consider a fluid moving through a pipe of
  nonuniform size (diameter)
• The particles move along streamlines in steady
  flow
• The mass that crosses A1 in some time interval is
  the same as the mass that crosses A2 in that same
  time interval

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Equation of Continuity, cont

• m1 m2 rA1v1 rA2v2
• Since the fluid is incompressible, r is a
  constant
• A1v1 A2v2
• This is called the equation of continuity for
  fluids
• The product of the area and the fluid speed at
  all points along a pipe is constant for an
  incompressible fluid

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Equation of Continuity, Implications

• The speed is high where the tube is constricted
  (small A)
• The speed is low where the tube is wide (large A)
• The product, Av, is called the volume flux or the
  flow rate
• Av constant is equivalent to saying the volume
  that enters one end of the tube in a given time
  interval equals the volume leaving the other end
  in the same time
• If no leaks are present

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Bernoullis Equation

• As a fluid moves through a region where its speed
  and/or elevation above the Earths surface
  changes, the pressure in the fluid varies with
  these changes
• The relationship between fluid speed, pressure
  and elevation was first derived by Daniel
  Bernoulli

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Bernoullis Equation, 2

• Consider the two shaded segments
• The volumes of both segments are equal
• The net work done on the segment is W (P1 P2)
  V
• Part of the work goes into changing the kinetic
  energy and some to changing the gravitational
  potential energy

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Bernoullis Equation, 3

• The change in kinetic energy
• DK ½ mv22 - ½ mv12
• There is no change in the kinetic energy of the
  unshaded portion since we are assuming streamline
  flow
• The masses are the same since the volumes are the
  same

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Bernoullis Equation, 4

• The change in gravitational potential energy
• DU mgy2 mgy1
• The work also equals the change in energy
• Combining
• W (P1 P2)V ½ mv22 - ½ mv12 mgy2 mgy1

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Bernoullis Equation, 5

• Rearranging and expressing in terms of density
• P1 ½ rv12 mgy1 P2 ½ rv22 mgy2
• This is Bernoullis Equation and is often
  expressed as
• P ½ rv 2 rgy constant
• When the fluid is at rest, this becomes P1 P2
  rgh which is consistent with the pressure
  variation with depth we found earlier

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Bernoullis Equation, Final

• The general behavior of pressure with speed is
  true even for gases
• As the speed increases, the pressure decreases

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Applications of Fluid Dynamics

• Streamline flow around a moving airplane wing
• Lift is the upward force on the wing from the air
• Drag is the resistance
• The lift depends on the speed of the airplane,
  the area of the wing, its curvature, and the
  angle between the wing and the horizontal

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Lift General

• In general, an object moving through a fluid
  experiences lift as a result of any effect that
  causes the fluid to change its direction as it
  flows past the object
• Some factors that influence lift are
• The shape of the object
• The objects orientation with respect to the
  fluid flow
• Any spinning of the object
• The texture of the objects surface

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Golf Ball

• The ball is given a rapid backspin
• The dimples increase friction
• Increases lift
• It travels farther than if it was not spinning

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Atomizer

• A stream of air passes over one end of an open
  tube
• The other end is immersed in a liquid
• The moving air reduces the pressure above the
  tube
• The fluid rises into the air stream
• The liquid is dispersed into a fine spray of
  droplets